calculus question!
How do you know when to take the ln of an equation when finding a derivative?

Question

calculus question!
How do you know when to take the ln of an equation when finding a derivative?

thatonegirl_ · Answer

For example... 
$$y=(sinx)^{x ^{2}}$$
My teacher said to take the ln of both sides first but how would u know when to do tht?

zepdrix · Answer

When you have a function raised to a function:$$\large\rm f(x)^{g(x)}$$When x is showing up in both the base and the exponent it creates a little bit of a problem.
That's when logs really really help.

thatonegirl_ · Answer

Oh okay thank you. So you couldn't just do chain rule for that?

satellite73 · Answer

"take the ln of both sides" lorda mercy

thatonegirl_ · Answer

Lol i forgot the name xD

satellite73 · Answer

no no that is fine 
the truth is that the definition of $$b^x$$is $$e^{x\ln(b)}$$ so the definition of $$(\sin(x))^{x^2}$$ is $$\huge e^{x^2\ln(\sin(x)}$$
and NOW you use the chain rule

zepdrix · Answer

Consider constant value n for these next two examples:
$\large\rm f(x)^n$ <-- simply apply power rule into chain rule

$\large\rm \frac{d}{dx}f(x)^n=nf(x)^{n-1}$

$\large\rm n^{f(x)}$ <-- apply exponential rule into chain rule

$\large\rm \frac{d}{dx}n^{f(x)}=n^{f(x)}(ln~n)f'(x)$

But when you x stuff in both the exponent and base, something else crazy is going on :)

zepdrix · Answer

Sure sure sure, or you can do that little clever trick that Sat mentioned ^

satellite73 · Answer

@zepdrix not really a trick 
other way is trick to get the variable out of the exponent
the way i wrote above appeals directly to the definition of an exponential

zepdrix · Answer

Woops I didn't apply chain rule in my power rule :x soz

thatonegirl_ · Answer

Ahh thank you both for the very thorough explanations :)

satellite73 · Answer

you will see that there is no free lunch, all the work entailf finding the derivative of $$x^2\ln(\sin(x))$$ using the product and chain rule